K.Alligood, J.Yorke, T.Sauer, Chaos - An introduction to dynamical systems, Springer Verlag, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for the Ikeda map T : R , de ned by T (x; y) x cos s y sin s; y cos s x sin s) where s = 1 x y ) 5:4; 0:9; 0:4; and = 0:92. The dynamics of T appear numerically to be chaotic in a region around the origin and to possess a chaotic attractor; see [1] for further details. We use GAIO to produce a covering of this attractor made up of 26863 boxes of equal size; see Figure 10. A transition matrix P on these 26836 boxes is produced using (3.4) and Figure 9 shows the 40 largest (in magnitude) eigenvalues of this 26863 26863 matrix. Note the ....

Kathleen Alligood, Tim Sauer, and James Yorke. Chaos: An Introduction to Dynamical Systems. Textbooks in Mathematical Sciences. Springer-Verlag, New York, 1997.

....analyze these kinds of cooperative coevolutionary algorithms. We then describe the particular CCEAs under study and show how MPS can be used to model them. In section four we present our initial MPS analysis, introduce the idea of using rain gauge measures as a form of empirical model validation [6], and then use this method to shed light of the behavior of a CCEA as an optimizer on two simple functions. Finally, we will conclude by discussing the impact of these results, both in terms of the specifics of what is learned, as well as the general message regarding the applicability of the MPS ....

Kathleen Alligood, Tim Sauer, and James Yorke. Chaos: An Introduction to Dynamical Systems. Springer--Verlag, 1996.

....there is a non linear transformation which makes it possible to entirely rebuild the matrix from d columns. To estimate the fractal dimension of the autoregressive matrix, we use the Grassberger and Procaccia method [4] many other methods can however be used to estimate a fractal dimension [1, 6, 7]. It must be mentioned that the concept itself of non linear dependency is difficult to define. Therefore the fractal dimension found by these methods can vary; in difficult situations, it may be worthwhile to use several methods in order to asses their results. The intrinsic dimension can also be ....

Alligood K. T., Sauer T. D., Yorke J. A.: Chaos: An Introduction to Dynamical Systems. Springer-Verlag, New York (1997), pp. 537-556

....From a dynamics view point, TCP deployed over wide area networks together with the routers represents a non linear dynamical system with delayed feedback. In general, such systems can be chaotic even under very simple formulations, and in fact it is less often that such system are non chaotic [2]. Furthermore, the feedback delay plays a crucial role in nonlinear systems [9] for ordinary di erential equations, a 1 dimensional system with delayed feedback can be chaotic, whereas without feedback it takes at least the dimensionality of 3 to generate chaos [24] TCP together with its ....

....network delays prolongs the duration of regime two, which when coupled with the background trac and small bu ers generates very complicated dynamics under certain conditions. We show that the dynamics of the window size updates embed a map which is qualitatively similar to the well known tent map [2] that generates chaotic trajectories. The former, however, is more complicated and has gaps in certain regions of the domain. Full analysis of this map will be presented in a forth coming paper. We adopt the following informal working de nition of chaos (more formal treatments can be found for ....

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K. T. Alligood, T. D. Sauer, and J. A. York. Chaos: An Introduction to Dynamical Systems. SpringerVerlag New York, Inc., 1997.

....phenomena termed chaotic have generally been studied in the context of dynamical systems. Dynamical systems are mathematical descriptions of physical systems that evolve deterministically in time. Good introductions to dynamical systems and chaos can be found in Devaney [15] and Alligood et al. [5]. In what follows we will consider a standard model for dynamical systems. In this model, the relevant states of the system form a compact subset of R . The time evolution of the system is described by an invertible map F : according to the following rule: if at time i the system is in ....

Alligood, K., Sauer,T., and Yorke, J.A., Chaos: An Introduction to Dynamical Systems, Springer, New York, 1996.

....for the Ikeda map T : R 2 , de ned by T (x; y) x cos s y sin s; y cos s x sin s) where s = 1 x 2 y 2 ) 5:4; 0:9; 0:4; and = 0:92. The dynamics of T appear numerically to be chaotic in a region around the origin and to possess a chaotic attractor; see [1] for further details. We use GAIO to produce a covering of this attractor made up of 26863 boxes of equal size; see Figure 10. A transition matrix P on these 26836 boxes is produced using (3.4) and Figure 9 shows the 40 largest (in magnitude) eigenvalues of this 26863 26863 matrix. Note the ....

Kathleen Alligood, Tim Sauer, and James Yorke. Chaos: An Introduction to Dynamical Systems. Textbooks in Mathematical Sciences. Springer-Verlag, New York, 1997.

....qui permet de reconstruire entirement la matrice partir de d colonnes. Il s agit donc bien de la dimension intrinsque recherche. Pour calculer cette dimension fractale, nous utiliserons la mthode de Grassberger et Procaccia [4] mais il en existe de nombreuses autres dans la littrature [1,6,7]. Le rsultat de cette mthode pouvant tre une valeur non entire, nous prendrons pour la suite la valeur entire la plus proche et nous l appellerons d (n tant la dimension d un vecteur initial) 2.3 Vecteur auto rgressif non linaire L tape suivante consiste construire un vecteur auto rgressif ....

K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (N-Y: Springer-Verlag, 1997), pp. 537-556.

....The locus is plotted in Fig. 2 for ease of reference. 3) The critical value of depends on the values of and . Fig. 3 shows the boundary curves where the sign of the real part of the complex eigenvalues changes. On these curves, the system loses stability via a supercritical Hopf Bifurcation [11] [13]. Remarks: To establish a supercritical Hopf bifurcation formally, one needs to show that, for given and , there exists for which the following conditions are satisfied by the complex eigenvalue pair [13] 30) 31) 32) where is the critical value of at which a supercritical Hopf bifurcation ....

.... On these curves, the system loses stability via a supercritical Hopf Bifurcation [11] 13] Remarks: To establish a supercritical Hopf bifurcation formally, one needs to show that, for given and , there exists for which the following conditions are satisfied by the complex eigenvalue pair [13]: 30) 31) 32) where is the critical value of at which a supercritical Hopf bifurcation occurs. Note that the last condition is necessary to ensure that the complex eigenvalue pair moves from the left side to the right side of the complex plane (preventing it from locusing along the ....

K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems. New York: Springer-Verlag, 1996.

....) x n x F (p# ) u#p] 12) In Eq. 12) the Jacobian matrix M is evaluated at the xed point x F (p# ) of the unperturbed system, which is the one to be stabilized. Since x F (p# ) is embedded in the chaotic attractor, it is unstable and it has one stable and one unstable direction [28]. Let e # and e # be the stable and unstable unit eigenvectors at x F (p# ) respectively, and let f # and f # be two unit vectors that satisfy f # ) e # f # ) e # 1 and f # ) e # f # ) e # 0, which are the relations by which the vectors f # and f # can be ....

K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, New York, 1997.

.... ) x 2 = x 1 Gamma x 2 x 3 x 3 = Gammax 2 (8) where OE(x 1 ) m 1 x 1 m 0 Gamma m 1 2 (j x 1 1 j Gamma j x 1 Gamma 1 j) This system is known to have a so called double scroll chaotic attractor for ff = 15:6, m 0 = Gamma 8 7 , m 1 = Gamma 5 7 , and 23 31 (see e.g. [1]) We assume that y = x 2 is the transmitted signal. Note that with this choice of y we have that the relative degree of y with respect to equals 2. Note further that, although it has been shown experimentally that the (x 1 ; x 3 ) subsystem synchronizes with the system ae x 1 = ....

Alligood, K.T., T.D. Sauer and J.A. Yorke, Chaos - An introduction to dynamical systems, Springer, New York, 1997.

....work one does not measure all components of the state vector. In this sense, the Henon map is an unrealistic model of experimental dynamics. We therefore examine two cases where the measurement captures only one component of a two dimensional state vector: the Ikeda map and the Tinkerbell Map [11]. The Ikeda map (with a complex valued state vector #x t ) #x t 1 = c 1 c 2 #x t exp i # c 3 c 4 (1 #x t 2 ) # z t = Re(#x t ) t with parameters c 1 = 1, c 2 = 0.7, c 3 = 0.4 and c 4 = 6.0. The Tinkerbell map operates on state (x t , y t ) x t 1 = x 2 t y 2 t ....

K.T. Alligood, T.D. Sauer, J.A. Yorke, "Chaos: An introduction to dynamical systems" Springer-Verlag 1996

....in R. This implies an infinity of homoclinic intersections and the existence of a chaotic orbit. The unstable eigenvector at L has a negative slope given by ( Gammaffi L = 1L ) Therefore it must have a heteroclinic intersection with SR . There is a mathematical result, called the Lambda Lemma [21], which says that if a curve C crosses a stable manifold transversally, then each point of the unstable manifold of the same saddle fixed point is a limit point of S n 0 f n (C) Since both UL and UR have transverse intersections with SR , by the Lambda Lemma we conclude that for each point ....

K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: an Introduction to Dynamical Systems. Springer, 1996.

.... sat( is the saturation function given by sat(w 1 ) 1 2 (j w 1 1 j j w 1 1 j) For the parameter values = 15:6, m 0 = 8 7 , m 1 = 5 7 , 25, this system is known to have a so called double scroll chaotic attractor, in which three unstable equilibrium points are embedded (see e.g. [1]) We assume that y = w 1 is the transmitted signal, so that (12) takes the form (9) Note that with this choice of y the pair (C; A) is observable. As the receiver, we take the system x = Ax (w 1 ) bu (13) where u 2 IR and b 2 IR n . Note that we now have that for a generic choice of b ....

.... C that is (not uniformly) exponentially attracting for all initial conditions w(0) 2 IR 2 f0g, i.e. 8w(0) 2 IR 2 f0g) 9 ; 0) 8t 0) d(C; t (w(0) e t ) where d(C; w) denotes the distance between C and the point w 2 IR 2 , and t ( denotes the ow of (14) Proof See e.g. [1]. The following result, that may be less well known, will be useful in the sequel. The proof of this result is based on modi cation of an argument in [14] Proposition 4.2 Consider the di erential equation (14) and let C denote its limit cycle. Let w(t) be a periodic solution that starts on ....

Alligood, K.T., T.D. Sauer and J.A. Yorke, Chaos - An introduction to dynamical systems, Springer, New York, 1997.

.... 1 ) x 2 = x 1 Gamma x 2 x 3 x 3 = Gammax 2 (7) where OE(x 1 ) m 1 x 1 m 0 Gamma m 1 2 (j x 1 1 j Gamma j x 1 Gamma 1 j) This system is known to have a so called double scroll chaotic attractor for ff = 15:6, m 0 = Gamma 8 7 , m 1 = Gamma 5 7 , and 23 31 (see e.g. [1]) We assume that y = x 2 is the transmitted signal. Note that in this case the relative degree of y with respect to equals 2. Further note that, although it has been shown experimentally that for constant the (x 1 ; x 3 ) subsystem synchronizes with the system ( x 1 = ff( Gamma x 1 x 2 ....

Alligood, K.T., T.D. Sauer and J.A. Yorke, Chaos - An introduction to dynamical systems, Springer, New York, 1997.

.... given by sat(w 1 ) 1 2 (j w 1 1 j Gamma j w 1 Gamma 1 j) For the parameter values ff = 15:6, m 0 = Gamma 8 7 , m 1 = Gamma 5 7 , fi = 25, this system is known to have a so called double scroll chaotic attractor, in which three unstable equilibrium points are embedded (see e.g. [1]) We assume that y = w 1 is the transmitted signal, so that (12) takes the form (9) Note that with this choice of y the pair (C; A) is observable. As the receiver, we take the system x = Ax Psi(w 1 ) bu (13) where u 2 IR and b 2 IR n . Note that we now have that for a generic choice of ....

....take a Van der Pol system of the form 8 : w 1 = w 2 w 2 = Gammaw 1 Gamma (w 2 1 Gamma 1)w 2 y = w 1 (14) The only equilibrium of this system is the origin, which is an unstable focus. Thus, the system (14) does not satisfy the hypotheses in ( 3] Further, it is well known (see e.g. [1]) that this system has a limit cycle C that is attracting for all initial points w(0) 2 IR 2 Gamma f0g. Consider a solution ( w 1 (t) w 2 (t) that starts on C, and let T denote its period. Define p(t) w 1 (t) 2 Gamma 1. By modifying an argument in [13] it may then be shown that ....

Alligood, K.T., T.D. Sauer and J.A. Yorke, Chaos - An introduction to dynamical systems, Springer, New York, 1997.

....6.3 Relationships with analysis of discretizations: continuation . 26 6.4 Relationship with the continuum random tree . 27 1 Motivation In recent years much attention has been given to computer implementations of dynamical systems with chaotic behavior [4, 10, 18]. In contrast to a continuous system, a computer implementation is a transformation on a finite, albeit rather large, set of digital numbers. Such a transformation may be viewed as a digraph on a set f1; 2; ng, with the property that each node is the initial node of exactly one edge. An ....

Alligood, K.T., Sauer, T.D., and Yorke, J.A. Chaos: An Introduction to Dynamical Systems. Springer Verlag, New York. 1997.

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K.Alligood, J.Yorke, T.Sauer, Chaos - An introduction to dynamical systems, Springer Verlag, 1995.

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ALLIGOOD K. T., SAUER T. D., YORKE J. A.: Chaos: An Introduction to Dynamical Systems. Springer, New York, 1996.

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K. T. Alligood, T. D. Sauer, and J. A. York, Chaos: An introduction to dynamical systems, Springer-Verlag, 1997.

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Kathleen T. Alligood, Tim D. Sauer, and James A. Yorke, Chaos: An introduction to dynamical systems, Springer-Verlag, Berlin, 1997.

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Kathleen Alligood, Tim Sauer, and James Yorke. Chaos: An Introduction to Dynamical Systems. Springer, 1996. Textbooks in Mathematical Sciences.

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K.Alligood, J.Yorke, T.Sauer, Chaos - An introduction to dynamical systems, Springer Verlag, 1995.

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K.T. Alligood, T.D. Sauer, J.A. Yorke, "Chaos: An introduction to dynamical systems" Springer-Verlag 1996

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K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: an introduction to dynamical systems, Springer-Verlag New York, Inc., 1996.

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Alligood, K.T., T.D. Sauer, and J.A. Yorke. (1997). Chaos: An Introduction to Dynamical Systems. New York: Springer-Verlag.

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Alligood, K.T., T.D. Sauer, and J.A. Yorke. (1997). Chaos: An Introduction to Dynamical Systems. New York: Springer-Verlag.

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